Method and apparatus for describing circular arcs



July 24, 1962 A. R. SPICACCI METHOD AND APPARATUS FOR DESCRIBINGCIRCULAR ARCS Filed March 28, 1960 INVENTOR.

Arrow 1 3,045,350 METHODAND APPARATUS FOR DESCRIBE? CIRCULAR ARCSAttilio R. Spicacci, Germantown Manor, Greene and Hortter Sts.,Philadelphia 19, Pa. Filed Mar. 28, 1960, Ser. No. 18,112 2 Claims. (Cl.33-27) This invention relates generally to a novel method and apparatusfor describing circular arcs.

As is well known to those versed in the art of mechanical drawing, it isoften necessary to describe arcs of relatively large radii beyond therange of ordinary drawing compasses. It has heretofore been necessary indrawing arcs of relatively great radii to employ a beam compass.However, the beam compass, when employed to draw arcs of large radii, isquite cumbersome, unwieldy, and otherwise difficult to use. Moreover,when the centerpoint of the arc is inaccessible, the beam compass cannotbe used, and approximations or tedious expediencies must be employed.

Accordingly, it is an object of the present invention to overcome thedifficulties mentioned above in describing arcs of large radii, and toprovide a method and apparatus which enables one to quickly, easily, andaccurately describe an arc of any desired radius, no matter how large,and wherein one practicing the invention operates entirely in the regionof the are being described.

It is another object of the present invention to provide a method andapparatus having the advantageous characteristics mentioned in thepreceding paragraph, which is extremely versatile so as to be capable ofsolv ing practically all geometric-construction problems that can bedone with a conventional compass, and with equal or improved facility.

It is a further object of the present invention to provide a drawinginstrument of the type described which is extremely simple inconstruction and use, so as to be durable and reliable throughout a longuseful life, and which can be economically manufactured for sale at areasonable price.

Other objects of the present invention will become apparent upon readingthe following specification and referring to the accompanying drawings,which form a material part of this disclosure.

The invention accordingly consists in thefeatures of construction, andcombinations and arrangements of elements and method steps, which willbe exemplified in the following description, and of which the scope willbe indicated by the appended claims.

In the drawings:

FIGURE 1 is a plan view showing a body part or trammel of the apparatusof the present invention;

FIGURE 2 is a sectional view taken substantially along the line 2-2 ofFIGURE 1, and illustrating a marking device in association therewith;

FIGURE 3 is a partial sectional view taken substantially along the line3-3 of FIGURE 1;

FIGURE 4 is an enlarged fragmentary plan view of the body part of FIGURE1;

FIGURE 5 is a plan view of another part, or abutment member of theinstant apparatus;

FIGURE 6 is a sectional view taken substantially along the line 66 ofFIGURE 5 FIGURE 7 is a plan view showing the body part or trammel andabutment members in operative relation with each other; and

FIGURE 8 is a diagrammatic illustration of the geometrical principlesinvolved in the instant invention.

Referring now more particularly to the drawings; and especially toFIGURES 1-4 thereof, a generally flat body part or trammel is generallydesignated 10, and is advantageously formed of relatively stiff, flat,transparent r 3,45,35@ Patented Ju y 1962 material, such as plastic orthe like. The body part or trammel 10 includes a pair of outwardly,generally oppositely extending legs 11. The legs 11 are arrangedsymmetrically about the center line of the body part 14 as the line 22.Thus, the legs 11 of body part ltlare substantially coplanar, and areformed on one side, the lower side as seen in FIGURE 1, with straightedges 12 which define therebetween a reentrant or included angle ahaving its vertex on the center line 22. The angle a is of less thanand, the line 2-2 defines the bisector of the angle. The straight edges12 terminate short of their juncture or vertex and are arcuately cutout, as at 13 adjacent to but spaced from the center line 22.Intermediate the arcuate cutouts 13, the body part 19 is provided" witha projecting portion or extension 14, which extends along the centerline 22 into the space within the angle a. Formed in the extension 14 isa through opening or hole 15, which may be of tapering or conicalconfiguration to facilitate engagement therethrough of a marking deviceor stylus. The opening or hole 15 is located on the center line orbisector 22 of the angle a and within the angle, being spaced from thestraight edges 12 or their extensions a predetermined distance. Ofcourse, the spacing of the hole 15 from the edges 12 or their extensionsis the distance or length of a line normal to the edges or theirextensions. It should also be noted that the hole 15 is located withinthe triangle bounded by the edges 12 and a line extending between thedistal ends of the edges.

The legs 11 of the body part 10 terminate at their outer ends in knifeedges 16, which edges extend in parallelism with each other and with thebisector line 2-2. In addition, there may be provided adjacent to one orboth edges 16 a suitable marking or line 17 extending normal to andterminating at the respective edge 16. The markings 17 are located sothat a line through the markings is normal to the center line orbisector 2-2. The stylus-receiving hole 15 is also located ona linebetween the markings 17.

Also provided on the body part, extending outwardly directly opposite tothe extension 14, is an extension 19. The extension 19, which extendsupward in FIGURE 1 along the center line or bisector 22, terminates atits upper or outer end in a knife edge 20 extending normal to andintersecting with the center line. Provided on the extension 19 is amarking 21 coincident with the center line or bisector 2-2 and which maybe a line extending normal to the edge 20.

In addition to the body part 10', there are provided a pair of discelements or abutment members 23, one being shown in FIGURES 5 and 6.Each abutment member includes a generally fiat, preferably transparentcircular body or disc 24, advantageously fabricated of plastic, andprovided with markings in the nature of intersecting lines or crosshairs 25 which intersect at the center of the circular body. A pair ofpins 26 are embedded in the body 24 of the disc element 2.3,eccentrically thereof, with their points projecting downward beneath theunderside of the body for impaiing in a drawing board or similarsupporting surface.

As seen in FIGURE 7, a pair of disc members 23 are provided. It shouldalso be noted that each of the disc members is of a radius equal to thedistance, as discussed hereinbefore, between the hole 15 and each of thestraight to the length of the chord 30 between the points 31aud 32, suchthat the function Zsin a is equal to the ratio of 3 the length of chord30 to R. Therefore, the length of chord or line 30 between points 31 and32 equals R 2 sin a.

After drawing the line or chord 30, one of the abutment members or discs23 is placed at each end of the chord with the intersection of its crosshairs 25 directly over the adjacent point 31,32. In this location, thepins or prongs 26 of each disc member 23 are impaled in the supportingsurface to anchor the disc members in position.

The body part is then placed flat on the supporting surface and itsstraight edges 12 brought into respective engagement with the circularor cylindrical peripheral edges of respective disc members 23, thecondition shown in FIGURE 7. A marking device, stylus, or pencil, as at36 in FIGURE 2, is inserted through the hole to engage the supportingsurface, and the body part or trammel 10 in the condition of FIGURE 7 ismoved with the edges 12 in sliding engagement with the respective discs23. Thus, the extension 14 and its received marking device move betweenthe disc members 23, and the are 33 is thereby marked. or traced on thesupporting surface. The traced are 33 will be circular and of the radiusR having the hereinbefore-mentioned mathematical relationship to theangle a and the length of chord 30.

This may be mathematically demonstrated (without rigor) in the followingmanner:

(1) In plane geometry, if the sides of a constant angle move throughfixed points, the locus of the vertex of the angle is a circular arc;and

(2) A chord of such are through the mentioned fixed points has a lengthequal to the radius of the are times twice the sine of the constantangle.

In the diagram of FIGURE 8, the sides 38 of the angle a respectivelypass through fixed points 39, and upon move ment of the sides 38 throughtheir respective fixed points 39 while maintaining the angle constantthe locus of points defined by the vertex 40 is a circular are 41 havinga radius R and a chord 42 extending between the points 39.

The ratio, chord 42zradius R=2 sin a.

Referring again to FIGURE 7, as the hole 15 of extension 14 is spacedfrom the extensions of straight edges 12 a distance equal to the radiusof each disc member 23, a straight line between each point 31, 32 andthe hole 15 is always parallel to the adjacent straight edge 12throughout the sliding motion of the body part 10. Therefore, an angledefined between the points 31, 15, and 32 is constant and equal to theangle a between the edges 12 throughout the sliding operation of thebody part 10.

If, for example, it is desired to determine the radius of are 33, andthe angle a is known to be l7249l0", the fixed ratio between the chordand radius R of the arc equals 4. While a single body part 10 of onepredetermined angle a is capable of use over a wide range of conditions,it may be desirable to provide a set of such body parts havingdiflferent fixed angles a for different chord-toradius ratios.

If it is desired to draw a tangent at any point to the are 33, say atthe point 34, it is only necessary to mark the locations of lines 17, atopposite ends of the legs 11, and then draw a tangent, as at 45 throughthe lines 17 and point 34. Also, a normal to the are 33 may be providedat any desired point, say point 34, by drawing a line through the latterpoint and through the marking line 21 of the extension 19, such a linebeing indicated at 46.

While certain operations employing the instant device and method havebeen described hereinbefore, it is ap parent that many additionaloperations may also be performed. For example, the following geometricalconstructions may be easily executed employing the method and apparatusof the instant invention:

(1) Extending an are on either end;

('2) Having a given arc, to describe another are of another radiustangent to the given are at a given point;

(3) To describe an arc of a given radius through two given points;

(4) To determine the radius of a given arc;

(5) To describe a circular are through three given points; and a (6) Todescribe an arc concentric with and at a given distance from a givenarc.

It will now be understood that the method and apparatus of the instantinvention is capable of performing a wide variety of geometricalconstructions, wherein it is entirely unnecessary to use the center ofan arc, and wherein the entire operation is performed in the region ofthe arc to be described without regard to the length of the radius ofthe arc.

In describing an arc according to the method of the present invention,the method may be considered as involving the following procedure:

(1) Providing a constant angle having a fixed relation to the ratio of achord to the radius of any circular are;

(2) Laying out a chord of a length determined by the particular ratio;

(3) Fixing the abutment members at the ends of the laid-out chord; and

4) Sliding the fixed angle along the abutment members to describe an arcdefined by the locus of the angle vertex.

The device and method are capable of extremely accurate results, as highaccuracy may be built into the body part, and there are no moving parts,as would require manufacturing tolerances and impair accuracy.

From the foregoing, it will now be appreciated that the presentinvention provides a method and apparatus for describing circular arcsand accomplishing related geometrical constructions, which fullyaccomplish their intended objects and are well adapted to meet practicalcon ditions of manufacture and use.

Although the present invention has been described in some detail by wayof illustration and example for purposes of clarity of understanding, itis understood that certain changes and modifications may be made withinthe spirit of the invention and scope of the appended claims.

For example, it is appreciated that the stylus receiver or opening 15may be located approximately at the vertex of the angle a, and that theabutment members 23 may then be slender pins or the like of minuteradius. While such variation is not preferred for accurate work, itmight be satisfactory for certain purposes, and of course iscontemplated as embraced within the instant invention.

What is claimed is:

1. A drawing instrument comprising a generally flat unitary body havinga pair of fixed straightedges defining therebetween an angle of lessthan and adapted to rest on a supporting surface with said edges in aplane parallel to said supporting surface, an extension on said body atthe juncture of said straight edges for locating a tracing point on thebisector of said angle at a predetermined distance from said straightedges, and a pair of abutment members each having its radius equal tosaid predetermined distance and adapted to be selectively positioned ona supporting surface, whereby said tracer point is adapted to describean are between the centers of said discs upon sliding movement of saidstraight edges along said discs, said body being provided at oppositeedges thereof with a pair of fixed tangent markings in alignment witheach other through the location of said tracer point for locating atangent to said arc, and said body having an additional edge providedwith a fixed normal marking coincident with the bisector of said angleextended externally of said angle.

2. A drawing instrument comprising an integral body having a pair ofstraight edges defining therebetween a fixed angle and adapted to reston a supporting surface with said straight edges in a plane parallel tosaid supporting surface, an extension on said body at the juncture ofsaid straight edges for locating a tracing point on the bisector of saidangle at a predetermined distance from said straight edges, and a pairof disc members each having its radius equal to said distance andadapted to be selectively positioned on a supporting surface, wherebysaid tracer point is adapted to describe an arc between centers of saiddisc members upon sliding movement of the said straight edges along saiddisc members, said disc members each being provided with a centralmarking for accurate location over a point on said surface, said bodybeing generally fiat and having a second pair of straight edges parallelto each other and to the bisector of said angle, there beingmarkings onsaid second pair of straight edges in alignment with each other throughthe location of said tracer point, and said body being formed with anadditional straight edge extending normal to and intersecting with thebisector of said angle extended externally of said angle, saidadditional straight edge being provided with a marking in alignmentwiththe external extension of said angle bisector.

References Cited in the file of this patent s FOREIGN PATENTS

